已知函数f(x)=x^2-2mx+m^2-m,g(x)=x^2-(4m-1)x+4m^2+mh(x)=4x^2-(12m+4)x+9m^2+8m+12令集合M＝{x|f(x)Xg(x)Xh(x)=0}且M为非空集合,求实数m的取值范围.

Question

已知函数f(x)=x^2-2mx+m^2-m,g(x)=x^2-(4m-1)x+4m^2+m h(x)=4x^2-(12m+4)x+9m^2+8m+12 令集合M＝{x|f(x)Xg(x)Xh(x)=0}且M为非空集合,求实数m的取值范围.

Answer 1

∵f(x)×g(x)×h(x) = 0,则三个因子至少有一个为零.1、当f(x) = 0时,即x�� - 2mx + m�� - m = 0△ = 4m�� - 4(m�� - m) ≥ 0m ≥ 0 2、当g(x) = 0时,即x�� - (4m - 1)x + 4m�� + m = 0△ = (4m - 1)�� - 4(4m�� + m) ≥ 0m ≤ 1/12 3、当h(x) = 0时,即4x�� - (12m + 4)x + 9m�� + 8m + 12 = 0△ = (12m + 4)�� - 16(9m�� + 8m + 12) ≥ 0m ≤ -11/2 故m的取值范围为m∈(-∞,-11/12)∪[0,1/12]